Special Issue – TANGO

Intrinsic thermoacoustic modes andtheir interplay with acoustic modesin a Rijke burner

N Hosseini1,2 , VN Kornilov1, I Lopez Arteaga1,3, W Polifke4,OJ Teerling2 and LPH de Goey1

Abstract

The interplays between acoustic and intrinsic modes in a model of a Rijke burner are revealed and their influence on the

prediction of thermoacoustic instabilities is demonstrated. To this end, the system is examined for a range of time delays,

temperature ratios and reflection coefficients as adjustable parameters. A linear acoustic network model is used and all

modes with frequency below the cut-on frequency for non-planar acoustic waves are considered. The results show that

when reflection coefficients are reduced, the presence of a pure ITA mode limits the reduction in the growth rate that

usually results from a reduction of the reflection coefficients. In certain conditions, the growth rates can even increase by

decreasing reflections. As the time delay of the flame and thus the ITA frequency decreases, the acoustic modes couple

to and subsequently decouple from the pure ITA modes. These effects cause the maximum growth rate to alternate

between the modes. This investigation draws a broad picture of acoustic and intrinsic modes, which is crucial to accurate

prediction and interpretation of thermoacoustic instabilities.

Keywords

Intrinsic thermoacoustic instability, Rijke burner, network model, acoustic mode, ITA mode

Date received: 19 May 2017; accepted: 30 April 2018

1. Introduction

Thermoacoustic instabilities constitute a major branchof combustion instabilities that are undesirable forcombustion systems designed to work steadily. Suchinstabilities are a result of a strong coupling betweenthe system acoustics and fluctuating heat generation.The acoustic fluctuations modulate the heat releaseand this in turn produces acoustic waves. A thermoa-coustically unstable system may exhibit pressure fluctu-ations with an amplitude in the order of audible sound(in e.g. heating appliances) up to operating pressurelevels (in e.g. rocket engines). These pressure fluctu-ations limit the operating range of the system or evencause structural damage and thus should be avoided orcontrolled.

The common practice in studying thermoacousticinstabilities is to first identify the coupling betweenthe active elements (such as burners and heaters) andthe acoustic field. Then it is possible to calculate thecomplex eigenfrequencies of the system including the

active and passive elements (such as ducts, area changesand terminations). This approach is used in the frame-work of the so-called network models and numericalHelmholtz solvers. In traditional thermoacoustics, it isunderstood that for a system to be thermoacousticallyunstable, a coupling should exist between the activeelements and the acoustic reflections from the rest ofthe system. However, it was established recently that a

1Mechanical Engineering Department, Eindhoven University of

Technology, Eindhoven, the Netherlands2Bekaert Combustion Technology BV, Assen, the Netherlands3Department of Aeronautical and Vehicle Engineering, KTH Royal

Institute of Technology, Stockholm, Sweden4Mechanical Engineering Department, Technische Universitat Munchen,

Garching, Germany

Corresponding author:

N Hosseini, Mechanical Engineering Department, Eindhoven University of

Technology, P.O. Box 513, 5600 MB Eindhoven, the Netherlands.

Email: naseh.hosseini@gmail.com

Creative Commons Non Commercial CC BY-NC: This article is distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 License (http://

www.creativecommons.org/licenses/by-nc/4.0/) which permits non-commercial use, reproduction and distribution of the work without further permission provided the

original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage).

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combustor may be unstable even in the absence of anyacoustical reflections upstream or downstream of theburner. In practice, such conditions have been observedas instabilities that do not respond adequately to thechanges in upstream and/or downstream acoustic prop-erties.1 This type of instabilities is thought to originatefrom an inherent coupling within the burner itself andare called intrinsic thermoacoustic (ITA) instabilities.

The modes of the coupled acoustic-flame systemtend to be associated either to an ‘‘acoustic mode’’ orto an ‘‘ITA mode’’, with the acoustic modes being asso-ciated to cavity acoustic oscillations and the ITA modesbeing associated to the inherent coupling of the burner.In addition, the system modes can be evaluated in twolimit cases that help investigating their properties. Thefirst case is when all elements in the system are con-sidered thermoacoustically passive, and the obtainedmodes are called ‘‘pure acoustic modes’’. The othercase is when the system boundaries or the boundariesat the interfaces of a selected element are considerednon-reflecting, and the obtained modes are called‘‘pure ITA modes’’.

Bomberg et al.2 have shown that when the flamescattering matrix is represented with respect to causal-ity, a pole of the transfer matrix can be found at aspecific frequency. The gains of the scattering matrixelements are maximum at this frequency, showingthat some kind of resonance is expected. They usedexperimentally measured transfer functions (TFs) of astable and an unstable burner to demonstrate the dif-ference in the thermal and acoustic responses of theflames. They identified the ITA feedback loop throughthe acoustics-flow-flame-acoustics coupling and wereable to relate the experimentally observed instabilityto the ITA instability. Hoeijmakers et al.3 theoreticallyshowed that the transfer matrix poles correspond to thefrequencies at which the phase of the flame/burner TFcrosses odd multiples of �. This means that, theoretic-ally, any active acoustic element that has a TF phase ofat least �, has at least one ITA mode. They also per-formed measurements with nearly anechoic conditionsupstream of the burner and showed that the trends ofthe observed instabilities are as can be expected fromtracking the poles of the transfer matrix. Therefore,they concluded that the observed instability was anITA mode. These two studies initiated a series of add-itional studies on the ITA instabilities in the past fewyears. Emmert et al.4 used a simplified network model,and Courtine et al.5 and Silva et al.6 used DNS to con-firm and capture ITA instabilities in confined laminarpremixed Bunsen flames.

The initial investigations on ITA instabilities havecreated the motivation and momentum for the commu-nity to investigate ITA instabilities in more detail.Consequently, recent studies have focused on studying

their behavior. Hoeijmakers et al.7 have numericallyshown that in practice a mode of the coupled acous-tic-flame system (where reflections are not perfect)could be a combination of pure acoustic and ITAmodes. In addition, Silva et al.8 recently reported meas-urements showing acoustic modes and ITA modesoccurring simultaneously in a swirl combustor.In order to make adequate mitigating choices it isvery important to identify if an unstable mode is acous-tic or ITA. In a recent study, Emmert et al.9 usedexperimentally measured values of flame TF within anetwork model and introduced a matrix transformationmethod that allowed them to decouple acoustic andITA modes from full system modes. By performing asweep on a defined coupling parameter, they observedthat some system modes originate from pure acousticand others from pure ITA modes. Their procedureidentifies the ITA modes, but the coupling parameterdoes not have a physical representation and reducing itto values below unity is unphysical. Albayrak et al.10

have very recently introduced a convective scaling forITA modes of a premixed swirl combustor. They haveshown that including flow inertia affects the frequenciesof ITA modes and the frequencies found using themodified approach better match with the experimen-tally observed unstable frequencies. These effects aresignificant in burners with large flow inertia, thus areachange. Their new criterion helps in associating thepeaks observed in the sound pressure level spectra toeither ITA or acoustic modes.

In a general scheme, the peculiarity of the problemwe consider is that an element with internal feedback isplaced inside a system with external feedback, and thensome interplay between these two feedback loops maybe expected. This situation is generic and may beencountered in different branches of science. One phys-ically close situation was studied in another branch ofacoustic research, which also may be interpreted interms of intrinsic and external feedback interplay. Theexample is the edge-tone configuration, which consistsof a planar jet from a thin slit that interacts with a thinsharp edge placed parallel to the slit further down-stream. The jet oscillations observed in this configur-ation are essentially a hydrodynamic instability.11

Moreover, it is well known that acoustical feedbackfrom surroundings influences the oscillation frequencyand amplitude. Some experiments have shown that theoscillation frequency of an edge-tone placed above atable was affected by the reflection of acoustic wavesfrom the table.11 It was also observed that surroundinggeometries affect the unstable frequency of the system.When the system is placed in the opening of a pipe, theedge-tone becomes an organ pipe and forms strongacoustical feedback. In such configurations, the edge-tone is not dominant; however, it plays a significant

2 International Journal of Spray and Combustion Dynamics 0(0)

role in the transients of the flue instruments.12,13

Additionally, Castellengo14 has shown that the edge-tone modes depend strongly on the geometry of themouth of the instrument. While for a recorder (wherethe jet length W is typically four times the jet thicknessh) only one edge-tone mode is active, for typical organpipes (where W/h can be as large as 12 or 20) manyedge-tone modes are observed. Therefore the attacktransient of an organ-pipe is often much more complexthan that of a recorder.

Most of the research on ITA instabilities has beenfocused on establishing a fundamental base for con-firming the existence and importance of ITA instabil-ities. In addition, using available experimental data toconfirm the findings has narrowed the window of inves-tigation to specific case studies. However, a broaderpicture of ITA and acoustic modes in a system isrequired in order to unveil more of the independentand coupled behavior of these modes. In addition,it is not always obvious whether a given mode is acous-tic or ITA, and whether any interplay exists betweenthese modes.

This work aims to address these issues by identifyingITA modes using multiple parametric sweeps. A Rijkeburner (a combustor with a flame and temperaturechange, but no area change) is investigated using alinear acoustic network model. All the system modeswith frequency below the cut-on frequency for non-planar acoustic waves are studied. This frequencyrange is usually below 1 kHz which is the range offocus for most thermoacoustic instability studies.The system is examined for a range of values offlame time delays, temperature ratios and reflectioncoefficients. Therefore, this work provides a paramet-ric study that governs a wide range of parameters;however, not every combination of parameters islikely to have some corresponding realistic configur-ation. Nevertheless, these parameters and their corres-ponding values are chosen based on (but not limitedto) the realistic conditions where thermoacousticinstabilities may occur.

The time delay and temperature ratio are directlyrelated to the combustion properties of the burner.A leaner flame results in larger time delays andsmaller temperature ratios. In addition, the use ofnon-conventional fuels can also result in various com-binations of flame time delays and temperature ratios.15

Moreover, a constant temperature ratio and varyingtime delay can mimic the cold start for perforated pre-mixed burners, since the stabilization temperature has alarge impact on the flame time delay.1 On the otherhand, the actual acoustic conditions at the boundariesare not ideal, due to the presence of other parts, and thereflection coefficients can vary. In addition, changingthe downstream temperature is meant to reveal

the effects of the shift in resonance frequencies andreducing the reflection coefficients towards zero toinclude the limits of anechoic environment. Therefore,the four mentioned parameters are chosen for the para-metric studies.

The results show complicated, yet interesting, inter-plays between the acoustic and ITA modes in thesystem. They reveal how an acoustic mode is affectedby a pure ITA mode and how these effects create largegrowth rates in multiple system modes. The movie sub-mitted as supplementary material very well visualizesthese effects and interplays. This investigation draws alarger picture of ITA instabilities and demonstratesthat such parametric investigations on multiple systemmodes are crucial to obtaining correct predictions andinterpretations of the instabilities in the system.

2. Theory of ITA instabilities

The approach in this article follows the one described inthe work of Emmert et al.16 The pressure fluctuations(p0) and velocity fluctuations (u0) together with theRiemann invariants f and g, are used to describe thepropagation of acoustic waves. The upstream (u) anddownstream (d) waves for the flame are illustrated inFigure 1, where u0 ¼ f-g and p0/�c¼ fþg.

For an acoustically compact flame (when its length isnegligible compared to the acoustic wavelength) theRankine-Hugoniot jump conditions17 can be appliedand the scattering matrix can be obtained. In lowMach number flows the upstream and downstreampressure and velocity perturbations are related as

p0d�dccu0d

24

35 ¼ " 0

0 1

� � p0u�ucuu0u

24

35þ 0

�TFu0u

� �ð1Þ

where TF is a velocity-sensitive transfer function asq0/q¼TF(u0/u) to describe the relation between velocityfluctuations (u0) and heat release fluctuations (q0).�¼ (Td/Tu)�1 denotes the dimensionless temperatureratio and "¼ (�ucu)/(�dcd) denotes the ratio of specificimpedances. These two ratios can be related based onsome assumptions. Since thermoacoustic instabilitiesusually occur in lean combustion of light hydrocarbons,such as methane, the number of moles remains almostconstant in the reactions, so the average molecular

p'uu'u

fu fdgdgu

p'du'd

Figure 1. The upstream (u) and downstream (d) waves for

the flame.

Hosseini et al. 3

weight does not considerably change. If the mixture isalso assumed a perfect gas then �¼ "2�1 relates thedimensionless temperature ratio to the ratio of specificimpedances.2,15

The feedback loop is well described by Bomberget al.2 and Albayrak et al.10 and shows that the intrinsicfeedback does not involve the acoustic reflections atthe combustor inlet and/or outlet. It is illustrated inFigure 2 with red colors showing that the upstreamtravelling wave (c2) generated by flame heat release,influences the velocity fluctuations upstream of theburner.

Respecting causality in defining the input and outputwaves leads to the scattering matrix as

fd

gu

� �¼ SM

fu

gd

� �ð2Þ

and using straightforward linear algebra we can obtain

SM ¼1

1þ TF "� 1ð Þ

2" 1þ �TFð Þ

"þ 1�1þ �TF� "

"þ 11þ �TF� "

"þ 1

2

"þ 1

2664

3775ð3Þ

The poles of the scattering matrix are obtained whenthe denominator in equation (3) equals zero, i.e. whenTF("�1)¼�1. These poles correspond to the pure ITAmodes of the system.2,7 Since " is real, this is only satis-fied when TF is also real. Therefore, the TF phase ’ ofpure ITA frequencies should equal odd multiples of �and gain is 1/("�1).

In order to close the coupling between the heatrelease and velocity fluctuations, the n-� model isused. This model is a ‘‘velocity-coupled’’ flame responsemodel and assumes that the unsteady heat release isproportional to the unsteady flow velocity, multipliedby the interaction index n, and delayed in time by

the time delay �.18 Therefore we have, TF¼ nei!�.The value of the time delay determines the change ofTF phase with frequency. It is important to note that nshows the ratio of the magnitude of the ‘‘fluctuations’’of the heat release to velocity, while � shows the‘‘mean’’ change of temperature, and thus, velocity.An acoustically passive flame has a zero n, but canhave a nonzero �. More detailed discussion of the roleand definition of the passive/active properties of theelements of a network in application to the thermoa-coustic instability analysis can be found in the work ofKornilov et al.19

If the time delay is assumed frequency-independent,the following expression can be realized for the pureITA mode of order m

2�fm ¼’m�¼

2m� 1ð Þ�

�) fm ¼

2m� 1

2�, m ¼ 1, 2, . . . ,

ð4Þ

where fm and ’m are the frequency and TF phase,respectively. Defining the Strouhal number as thenon-dimensional frequency, we have

St ¼ fm� ¼ m� 0:5, m ¼ 1, 2, . . . , ð5Þ

The next section contains the description of theRijke burner setup and its corresponding acoustic net-work model, which is chosen to investigate the behaviorof the acoustic and ITA modes.

3. The Rijke burner and its equivalentacoustic network model

An illustration of the Rijke burner showing the dimen-sions, location of the flame, upstream and downstreamtemperatures, and assumed flow direction is presentedin Figure 3. In this study, the flow is not modeled sinceits Mach number is small and its acoustic effects arenegligible. In addition, the flame is compact and itsthickness is negligible in comparison with the tubelength.

Figure 4 contains the equivalent acoustic networkmodel of the Rijke burner. Here the only active elementis the flame and the rest of the system are passive elem-ents (ducts and terminations).

Figure 2. The internal intrinsic feedback loop in a combustor.

R and T denote the reflection and transmission coefficients

upstream and downstream of the burner (recreated from

Albayrak et al.10).

Figure 3. The Rijke burner model showing the combustor

length L, diameter D, flame location X, upstream temperature Tu,

downstream temperature Td and assumed flow direction.

4 International Journal of Spray and Combustion Dynamics 0(0)

The dimensions and values used in the current inves-tigation are summarized as L¼ 1m, X¼ 0.25m,Tu¼ 300K, Td2[450:150:1800]K, �2[0.5:0.5:5], n¼ 1,�2[0:0.1:5]ms and R2[�1:0.2:0]. The tube diameter Dneeds to be small enough (in comparison with a cut-offdiameter of the lowest frequency of propagation oftransversal modes) to limit the system to only one-dimensional propagation of longitudinal acousticwaves. The fluid is considered air with temperature-dependent density and speed of sound. The flametime delay � covers a large range of values encounteredin academic cases, such as heated wires and laboratoryBunsen flames, as well as practical combustion systems,such as gas turbines and heating appliances.The dimensionless temperature ratio � is calculated as�¼ (Td/Tu)�1. Finally, the reflection coefficients at theupstream and downstream ends are varied from �1(fully reflecting open end) to 0 (non-reflecting), andare assumed equal, real and frequency-independent.

The constructed model is solved using taX,16 a state-space network modelling tool developed by TechnicalUniversity of Munich. This network model is a loworder simulation tool for the propagation of acousticwaves in acoustic network systems and is able to deter-mine the stability, mode shapes, frequency responses,scattering matrices and stability potentiality of theacoustic networks of arbitrary topology. The paramet-ric studies create 3060 cases, for which all the systemmodes within the frequency range of interest are inves-tigated. This broad investigation is expected to revealdetails of the behavior of the acoustic and ITA modesas well as their possible interplays. In the followingsections, the results are presented predominantly inthe form of pole-zero plots showing the eigenfrequen-cies and the corresponding growth rates of the systemmodes.

4. Results and discussion

The figures presented in this section show all the systemmodes in the frequency window of interest. The com-plete sweep on time delay is performed with steps of0.1ms. However, the data are presented only for a fewselected values, i.e. with steps of 0.5ms. For the com-plete version and better impression of the trends, thereader is encouraged to see the video file submitted assupplementary material, because the high resolutionanimation shows the ‘‘motion’’ of the modes. Eachframe of the animation shows the data for a specific

time delay. Therefore, using the Strouhal numberwould mean to multiply the frequency data by thevalue of time delay in every frame. In addition, theStrouhal number changes as the time delay increasesand this makes interpretation of the graphs more com-plicated. This is why the data are presented in terms ofabsolute frequency for this set of graphs. The center ofeach circle shows the frequency and growth rate of asystem mode, and the radius scales with respect to thetemperature ratio. The colors show the change in thereflection coefficients according to the included colormap. Therefore, the black circles represent the pureITA modes of the system. In the next two subsections,the results are split in groups of small (� < 1ms) andlarge (�� 1ms) time delays. The reference for this cat-egorization is the practical range encountered in aca-demic and industrial burners. Since the time delay isrelated to the convective time of perturbations in theflame, small flames usually have smaller time delaysthan large ones. The time delay of small laminar pre-mixed perforated burners is usually below 2ms, but forlarge turbulent swirl flames in gas turbines, it can be aslarge as 7ms.

4.1 Small time delays (0�s<1 ms)

The data for �¼ 0, 0.5 and 0.8ms are presented in Figure5. When �¼ 0 ms (Figure 5(a)), no instability is expectedand all system modes are stable, i.e. the growth rate isnegative. Here, decreasing the reflection coefficients(increasing acoustic losses) only further stabilizes thesystemmodes. This is the well-known behavior of acous-tic modes. In addition, it is easy to distinguish the first,second and third modes of the system aroundthe frequencies of 250, 550 and 900Hz, respectively,along with the fourth mode for the cases with small tem-perature ratios around the same frequencies as thethird mode (the sweep on the temperature ratio causesoverlaps in the frequency ranges of some modes). Byincreasing � from 0 to 0.5ms, the first and third modestart to destabilize, while the secondmode further stabil-izes. This result is expected based on the location of theflame and has been observed experimentally as well.20

However, as � approaches 0.5ms (see the supplementaryvideo file), the growth rate of the third mode increases toone order ofmagnitude larger than that of the firstmode.When �¼ 0.5ms, the first pure ITA mode enters thewindow of investigation and it is revealed that the thirdmode is significantly affected by the proximity of this

Reflective end

R

Duct

X, Tu

Flame

n-tDuct

L-X, Td

Reflective end

R

Figure 4. The equivalent acoustic network model of the Rijke burner.

Hosseini et al. 5

pure ITAmode. Therefore, the third mode which was anacoustic mode for smaller values of � is now an ITAmode. Therefore, the system now has two acoustic andone ITA modes in the investigated frequency range. Byslightly increasing the time delay to �¼ 0.8ms (Figure5(c)), a new acoustic mode enters the window of investi-gation following the first pure ITA mode, so that thesystem always has its three acoustic modes. This recon-firms the conclusions of Emmert et al.9 about the totalnumber of system modes being equal to the summationof acoustic and pure ITA modes.

In addition, decreasing reflection coefficients when�¼ 0.5ms makes the first and second mode morestable, while the third mode converges to the firstpure ITA mode, which is unstable for � > 3. In practice,this means that in a Rijke burner that has a flame with asmall time delay, increasing the mean heat release (byusing higher powers for example) can shift the instabil-ity from the first to the third mode. This depends alsoon the damping of these modes and the frequencydependence of the reflection coefficient, which in turnincreases with combustor diameter. Nevertheless, simi-lar effects have been observed experimentally before,but are not taken into account in this study.

In order to take a closer look at the ‘‘convergence’’of system modes to pure ITA modes, a finer sweep is

performed on the reflection coefficients between �0.2and 0. The results for the third mode are plotted inFigure 6. Here it can be seen in more detail that thecontinuous convergence to pure ITA modes occurs onlywhen �� 3.5. It is important not to mistake this limitwith the critical gain criterion in pure ITA modes.This criterion includes a critical value of the gain ofthe flame TF, above which the growth rate of thepure ITA mode is positive. It has been derived analyt-ically in previous works.4,5,7 For the investigated

-250

-200

-150

-100

-50

0

50

100

150

200

0 100 200 300 400 500 600 700 800 900 1000

]s/1[

etaR

htw

orG

Eigenfrequency [Hz]

= 0.8 ms

= 0.5 5.0

-1.0 -0.8 -0.6 -0.4 -0.2 0.0R=

qt

(a) (b)

(c)

Figure 5. The eigenfrequencies and growth rates of all system modes when �¼ 0 (a), 0.5 (b) and 0.8 ms (c), �2[0.5:0.5:5] (scaled by

marker size) and R2[�1:0.2:0] (scaled by color).

Figure 6. The eigenfrequencies and growth rates of all system

modes when �¼ 0.5 ms, �2[0.5:0.5:5] (scaled by marker size) and

R2[�0.2:0.04:0] (scaled by color).

6 International Journal of Spray and Combustion Dynamics 0(0)

configuration, the critical gain of n¼ 1 occurs at �¼ 3,which is in agreement with the theory.5

If the system is operating in �� 3.5 conditions,increasing acoustic losses does not fully stabilize thesystem. These modes are known as ITA modes.9 Inaddition, Figure 6 demonstrates that as the reflectionsdecrease, the growth rates of the ITA modes can changenon-monotonically (follow the line of reducing reflec-tions for �¼ 3.5). It can also be deduced that in thepresence of very small reflections, increasing � from 3to 3.5 dramatically changes the growth rate and a split-ting occurs in the trend due to the presence of a pureITA mode close to this frequency.

4.2 Large time delays (s�1 ms)

The eigenfrequencies and growth rates of all systemmodes for �¼ 1 and 1.5ms are plotted in Figure 7.As the time delay increases to 1ms (Figure 7(a)),the frequency of the first pure ITA mode reduces toaround 500Hz and the distribution of the systemmodes becomes more complicated. As the pure ITAmode moves to lower frequencies, it ‘‘drags’’ theITA mode with it and pushes the neighboring acousticmode (the second mode) towards lower frequencies.In Figure 7(a), it is visible that the third mode nowhas a frequency around 600 Hz, while it was around800Hz for �� 0.5ms, and it is still an ITA mode (i.e. itconverges to the first pure ITA mode as the reflectionsdecrease). On the contrary, other modes are acousticmodes and stabilize as the reflections decrease.

When �¼ 1.5ms (Figure 7(b)), the third mode isswitched back to an acoustic mode and the first pureITA mode is now affecting the first mode. Here, a‘‘movement’’ convention is chosen to describe theseinterplays (the reader is encouraged to follow this pro-cess with the supplementary video file). As a pure ITAmode passes an acoustic mode, it attaches to the acous-tic mode, turning it into an ITA mode (if the

temperature ratio is large enough), and then drags italong towards lower frequencies. The pure ITA modeeventually detaches from the ITA mode (causing it tobecome acoustic again) only to immediately attach toanother lower acoustic mode. This ‘‘attachment’’ and‘‘detachment’’ occur around the frequency of the third(900 Hz) and second (500Hz) pure acoustic modes,respectively. This occurs in such a way that the totalnumber of modes is equal to the summation of thenumber of pure acoustic and pure ITA modes.This has been observed for a specific set of system par-ameters in a previous study,9 but here it is shown that itholds for almost any combination of the governingparameters.

When �¼ 1.5ms (Figure 7(b)), the second pure ITAmode has the frequency of about 1 kHz and creates thesame effect as the first pure ITA mode when the timedelay was 0.5ms (Figure 5(b)). This behavior repeats asnew pure ITA modes enter the window of investigationand is verywell illustrated in the supplementary video fileand Figure 8. It is obvious that increasing the time delayresults in more and more pure ITA modes to enter thewindow of investigation. However, it can be observedthat new acoustic modes also enter the window of inves-tigation, following the pure ITA modes. The interplaybetween the acoustic and pure ITA modes of thesystem is complicated, but repetitive. The pure ITAmode holds to the system ITA mode if there are nolower acoustic modes to which it can attach. Therefore,they remain stitched together as a pack of ITA modes.This is visible for the first mode in Figure 8.

The results are so far presented in the form of snap-shots of the supplementary video file at specific timedelays which includes the whole parameter range.In order to better study how acoustic modes areaffected by pure ITA modes, it is efficient to choose aspecific temperature ratio and reflection coefficient andinvestigate the trends in more detail. This analysis ispresented in the next section.

(a) (b)

Figure 7. The eigenfrequencies and growth rates of all system modes when �¼ 1 (a) and 1.5 ms (b), �2[0.5:0.5:5] (scaled by marker

size) and R2[�1:0.2:0] (scaled by color).

Hosseini et al. 7

4.3 The behavior of system modes near pureITA modes

Here, the temperature ratio �¼ 5 and reflection coeffi-cient R¼�1 are chosen and the system modes are eval-uated in terms of growth rate and Strouhal numberSt¼f�. These values correspond to combustors withlarge temperature ratio and open inlet and outlet.Usually, high growth rates are expected for these con-ditions, which are expected based on the absence ofacoustic energy losses at the boundaries and the contri-bution of the temperature ratio to the acoustic energygain. In addition, the time delay is further increased to10ms in order to evaluate the system behavior for largeStrouhal numbers.

Figure 9 illustrates how the Strouhal numbers of thefirst six system modes change with the time delay.The fixed Strouhal numbers of the pure ITA modesare illustrated with horizontal dotted lines with theirvalues corresponding to equation (5), and the conver-gence of the first, second, and third system modes totheir corresponding pure ITA modes is visible. If nopure ITA modes were present, one could expect thatthe Strouhal number uniformly increases with timedelay. However, this figure clearly illustrates the attach-ing and detaching of the acoustic modes to pure ITAmodes in the form of limiting the increase of Strouhalnumber with time delay when the Strouhal number ofthe mode is close to that of a pure ITA mode. Thiseffect is less significant for higher modes and lower

Figure 8. The eigenfrequencies and growth rates of all system modes when �2[2:0.5:4.5] ms, �2[0.5:0.5:5] (scaled by marker size)

and R2[�1:0.2:0] (scaled by color).

8 International Journal of Spray and Combustion Dynamics 0(0)

pure ITA modes, but increases for higher pure ITAmodes.

The slope of the curve corresponding to a specificsystem mode in Figure 9 may be used as a criterionto define whether the mode is acoustic or ITA. Onemay define the system mode ITA, i.e. the mode isaffected by the presence of the pure ITA mode, whenthis slope is smaller than a certain value (it is zero forpure ITA modes).

The complete picture of the mode behavior isobtained when the growth rates are also included.Figure 10 illustrates the Strouhal numbers and growthrates of the first six system modes (indicated with Latinnumbers) for the investigated range of parameters (eachmarker indicates a system mode). Distinct fluctuationsbetween stability and instability are visible for all themodes and these fluctuations have a counter behaviorfor the even and uneven modes. This behavior isexpected based on the combination of � and the locationof the heat source. These fluctuations are also visible in

the animated video file submitted as supplementarymaterial. The maxima and minima occur very close tothe Strouhal number of the pure ITA modes; however,they deviate from it as the Strouhal number increases.Moreover, all system modes have zero growth rate forinteger Strouhal numbers.

In addition, Figure 10 shows that higher ordermodes have, in general, larger growth rates and theirmaximum growth rate decreases as Strouhal numberincreases. If the growth rates of these modes are largeenough, they may overcome their acoustic dampingand take over the system instability from the usualfirst-mode instability in a Rijke burner. Therefore, itis important to investigate higher order modes to beable to see such differences.

Investigating the system modes for various ranges ofparameters provides a large picture of the system modesand reveals the complex interplay between the acousticand intrinsic modes. Although the investigation is per-formed on a Rijke burner, the revealed interplays areexpected to be similar in more practical configurationswhen only longitudinal acoustic modes are considered.Therefore, the conclusions drawn in the next section aregeneric in nature and can be extended to other particularcases.

5. Conclusions

The thermoacoustic instabilities in a Rijke burner areinvestigated focusing on the ITA modes and their inter-play with the acoustic modes. A linear network modelis used and multiple parametric studies are performedwith varying values of time delay, temperature ratio,and reflection coefficients, covering a broad range cor-responding to various (sometimes extreme) operatingconditions of a combustor. This parametric studymimics practical situations, such as cold start (almostconstant temperature and varying time delay) andchanges in fuel composition (slowly varying tempera-ture ratio and quickly varying time delay).

The results show that the pure ITA and acousticmodes constantly interplay with each other, even forsmall values of flame time delay. While the commonbelief for ITA modes to be effective is that the timedelay needs to be relatively large, this investigationshows that ITA modes may be specifically responsiblefor the highest growth rates when time delay is rela-tively small (around 0.5ms). In addition, the maximumgrowth rate occurs in different modes for differentvalues of time delay. Therefore, the complete effectsof ITA instabilities are better captured if the investiga-tion is not limited to specific values of time delay or thefirst acoustic mode.

When reflection coefficients are reduced, the presenceof a pure ITA mode near an acoustic mode limits the

Figure 9. Changes in the Strouhal number of the first six

system modes with time delay. The mode number is shown in

Latin numbers, �¼ 5 and R¼�1.

Figure 10. The first six system modes (indicated with Latin

numbers) when �¼ 5, R¼�1 and �2[0:0.1:10] ms (each marker

indicates a system mode for a specific value of �).

Hosseini et al. 9

reduction in the growth rate of the mode to that of thepure ITA mode. In certain conditions, the growth rate ofthis mode increases by decreasing reflections. This meansthat a system that has an unstable ITA mode mightbecome even more unstable if acoustic damping is intro-duced at the reflective ends. Therefore, defining the char-acteristics of the instability by multiple parametricstudies is useful before an abatement method is selected.

As the time delay increases, the number of pureITA modes also increases and creates complicatedmode overlapping. However, the observed behaviorof the system modes with increasing time delay is peri-odic in nature, i.e. system modes couple to and subse-quently decouple from pure ITA modes and becomestable and unstable. This phenomenon is also seen inthe form of an ITA mode attaching to a pure ITA modefor certain time delays, and then detaching from it andbecoming an acoustic mode again. This motivates finerparametric investigations when thermoacousticinstabilities are investigated in a real setup in order toidentify the details of the occurred mode. In addition,this observation may be a potential cause of modeswitching in situations with varying time delay, suchas cold start.

In this investigation, interplays are revealed betweenthe acoustic and intrinsic modes in the system. Thebehaviors of these interplays are studied and it isdemonstrated that for a correct prediction of thermo-acoustic instabilities, it is crucial to create a broad pic-ture of system behavior showing how various systemmodes react to the changes of effective parameters.Moreover, the observed effects have similarities withother systems in other branches of physics that motiv-ates a future multidisciplinary holistic study on the sys-tems with internal and external feedback in order toreveal the common underlying mathematics.

Acknowledgements

The presented work is part of the Marie Curie Initial TrainingNetwork, Thermo-acoustic and aero-acoustic nonlinearitiesin green combustors with orifice structures (TANGO).

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.

Funding

The author(s) disclosed receipt of the following financialsupport for the research, authorship, and/or publication of

this article: European Commission under call FP7-PEOPLE-ITN-2012.

ORCID iD

N Hosseini http://orcid.org/0000-0003-0998-4977.

Supplement Material

Supplement material for this article is available online at

https://youtu.be/2lIcAViB3mM

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